Flop Combinatorics

I noticed last night that I was seeing a huge number of two-tone flops

I had no idea what the real number should be, nor could I find it on line.

I therefore decided to figure it out.

I turns out that there are 19,600 possible flops. Order of cards is disregarded -- ie,
is the same as

I used Excel's combin(number,number_chosen) function for this, and in this case that would be C(50,3)
50 because you hold two cards, so only 50 are unknown.
It turns out that the percentages are slightly different if you hold an offsuit hand compared to the percentage when you hold a suited hand.

Following this I checked my session, and I had 4 more 2-suited flops that the math would predict.
I'm either *extremely* perceptive or it is just another example of seeing patterns where there are none.

I'm either *extremely* perceptive or it is just another example of seeing patterns where there are none.

VS

Apparently seeing patterns is an evolutionary essential trait, while seeing erroneous patterns hardly affects our survival, but missing a tiger in the tall grass only once is instant gene pool removal.

It happens a lot to poker players. That is how they develop "lucky" hands, and "AQo always loses"

It happens at an unconscious (OK, subconscious
) level with a great negative effect.
If (for example) you lose the first several times you open-raise suited connectors, you may without thinking begin to remove them from your range.

I flopped quads twice last week.
Both times, my pocket pair hit the same pair on the flop.

I'll admit that I found the math daunting, so I did what any other lazy person would do:
I just counted 'em.
I used a Python script to generate all the possible flops, then went through the list and built the following tables.

I'm pretty sure this won't help anyone increase their winrate, but it is interesting...

The number for trips looked odd, but if you do not have a pocket pair, then there are 11 ranks with 4 combos of trips, and two with 1 each.

If you hold a pocket pair, then there are 12 ranks with 4 combos of trips each, and one rank with none, for 48.

This made me realize that all of the above are wrong if you have an offsuit pocket pair.
They're very wrong if you hold a suited pocket pair, actually.

If you do hold a pocket pair, then of the 50 unknown cards, two match your pair, leaving 48. Each of those 48 can go with your pocket pair 1 way, so you any time you hold a pocket pair, you have a 48/19600 or about 0.24% chance of hitting quads.